Norm Behavior Of Solutions To A Parabolic-elliptic System Modeling Chemotaxis In Two Space Dimension

Mathematical Biology mainly aims at ecological system, one specific property of ecological system is that the body in that can catch the external stimulate and response to that, this property can be called Taxis. As we known, there are many different types of taxis such as: aerotaxis, chemotaxis, geotaxis and haptotaxis and so on. This paper mainly introduce chemotaxis which is widely interested by many mathematicals and biologists. The chemotaxis movement is the body move towards the chemicals which are good for them and away from the chemicals which are bad for them. In fact, it is not only in animal and insect ecology that chemotaxis is important, it can be equally crucial in biological process where there are numerous examples. For example when a bacterial infection invades the body it may be attacked by movement of cells towards the sourcse as a result of chemotaxis. Convincing evidence suggests that leukocyte cells in the blood more towards a region of bacterial inflammation to counter it, by moving up a chemical gradient caused by the infection. So we can say that chemotaxis model has close relative with medicine, study of the model has great active meaning in the medicine.A widely studied chemotactic phenomenon is that exhibited by the slime mold Dictyostehum Discoideum where single cell amoebae more towards regions of relatively high concentrations of a chemical called cyclic-AMP which is produced by the amoebae themselves. So far many papers have deeply studied this model, under some special condations, there is a critical number , when the solution of this model exists globally, and if the solution will blow-up in finite time. The paper mainly consider the case ofε= 0 and reaction termαu—βv and the model is showed in (1.3). This paper mainly studied the relative property of the K—S model: the local existence, global existence, regulity and the blow-up property to model (1.3). First the paper discuss the local existence of solutions to the equivalent form of model (1.3) by use of the relative property of analytic semigroup and sectorial operator, then discuss the regularity of solutions. Meanwhile, we get the non-negative property of u and v. In the latter section the case of aα≥bβwere studied by use of two order moment and the estimate of two order moment, so we conclude that although reaction terms effect K—S model by a degree, but the critical number of system also exists.